If `y=sqrt(asqrt(asqrt(asqrt(a))))`, Then value of y?

Options

A. `a^(1/16)`
B. `a^(15/16)`
C. `a^(31/32)`
D. `a^(15/16)`

Answer

To solve the expression \( y = \sqrt{\sqrt{\sqrt{\sqrt{a}}}} \), we can break it down step by step.

### Step 1: Rewrite the square roots in exponent form
The square root can be expressed as an exponent of \( \frac{1}{2} \). Thus, we can rewrite the expression as:
\[ y = \sqrt{\sqrt{\sqrt{\sqrt{a}}}} = a^{\frac{1}{2}} \text{ (first square root)} \]

### Step 2: Apply the second square root
Now we take the square root of \( a^{\frac{1}{2}} \):
\[ y = \sqrt{a^{\frac{1}{2}}} = (a^{\frac{1}{2}})^{\frac{1}{2}} = a^{\frac{1}{4}} \text{ (second square root)} \]

### Step 3: Apply the third square root
Next, we take the square root of \( a^{\frac{1}{4}} \):
\[ y = \sqrt{a^{\frac{1}{4}}} = (a^{\frac{1}{4}})^{\frac{1}{2}} = a^{\frac{1}{8}} \text{ (third square root)} \]

### Step 4: Apply the fourth square root
Finally, we take the square root of \( a^{\frac{1}{8}} \):
\[ y = \sqrt{a^{\frac{1}{8}}} = (a^{\frac{1}{8}})^{\frac{1}{2}} = a^{\frac{1}{16}} \text{ (fourth square root)} \]

### Final Result
Thus, the value of \( y \) is:
\[ y = a^{\frac{1}{16}} \]